We can directly compute the Mellin transform of (REF ) and verify (REF ).
Taking this transform we compute
\(H_{1}(s) = \frac{C\,\Gamma (s)}{\pi } \int _{0}^{\infty } dr\; r^{-s} \sin \left( \frac{\pi }{\Gamma (\gamma )}r^{\gamma } \right) e^{-\cos (\pi \gamma )\Gamma (1-\gamma )r^{\gamma }} .\) 
under the condition \( \Re (s) < 1+\gamma \)  and the more restrictive \( \Re (\gamma ) < \frac{1}{2} \) , (\( \cos (\pi \gamma ) > 0 \) ).
Under an internal change of variable this integral becomes a known one, [1]}[pg. 490, 3.944, # 5], with an evaluation leading to (REF ).
In addition we see that \( C=1 \) .
